Page 50

var('a,b')
#  We declare to the program that symbols 'a' and 'b' are variables.

(3*a*(2*b+4)+3*b*(2*a+5)).expand() # Multiplication has to be written explicitly.

12*a*b + 12*a + 15*b

var('a,b')
#  We declare to the program that symbols 'a' and 'b' are variables.

show((3*a*(2*b+4)+3*b*(2*a+5)).expand()) # Better notation

$$\newcommand{\Bold}[1]{\mathbf{#1}}12 \, a b + 12 \, a + 15 \, b$$

Page 56

show((6661029375).factor())

$$\newcommand{\Bold}[1]{\mathbf{#1}}3^{2} \cdot 5^{4} \cdot 7^{2} \cdot 11 \cdot 13^{3}$$

Page 59

gcd(gcd(135,2520),300)

15

lcm(lcm(135,2520),300)

37800

Page 123

var('m,n,k')
show(((6*m^3*n^3*k+4*m^2*n^4)/(2*m^3*n^3)).simplify_full())

$$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{3 \, k m + 2 \, n}{m}$$

Page 154

solve((1+2*(4-x))/2-(2-x)/3==(6-2*x)/4+5/2,x)  # The sign for equality in the equation is written as ==!

[x == -1]

Page 161

var('R,R1,R2')
show(solve(1/R==1/R1+1/R2,R2))

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left[R_{2} = -\frac{R R_{1}}{R - R_{1}}\right]$$

Page 170

var('y')
solve([-3*x+5*y==8,x+7*y==6],x,y)

[[x == -1, y == 1]]

Page 171

var('s1,s2,t,v1,v2,d')
show(solve([s1==v1*t,s2==v2*t,d==s1+s2],s1,s2,t))

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[s_{1} = \frac{d v_{1}}{v_{1} + v_{2}}, s_{2} = \frac{d v_{2}}{v_{1} + v_{2}}, t = \frac{d}{v_{1} + v_{2}}\right]\right]$$

Page 222

Rational(0.25), (2/3).n(), Rational(2/3.n()), (Rational(0.25)).n()

(1/4, 0.666666666666667, 2/3, 0.250000000000000)

Page 260

print(pi.n())  # We calculate number pi to SageMath default precision.
print(pi.n(digits = 50)) # We calculate number pi to 50 significant digits.
print(round(pi,5))  # We round number pi to the fifth decimal.

3.14159265358979
3.1415926535897932384626433832795028841971693993751
3.14159

Page 281

var('t')
plot(10*t-5*t^2,(t,0,3), ymin = 0, aspect_ratio=1)
# First input in the command is the function we are plotting, 
# second input tells us the interval on which we plot it,
# the remaining two inputs are optional,
#- the first sets the lower limit to value plotting,
# since we are not interested in negative heights,
#- the second says that the same units are on both axes.

Page283

plot(x^3-x,(x,-2,2),ymin = -2, ymax=2, aspect_ratio = 1)

Page 284

var('y')
implicit_plot(x^2+y^2==1, (x,-2,2),(y,-2,2), aspect_ratio = 1, axes = true,frame = false)
# The first two inputs tell the software which equation we are plotting
# and in which area of the plane.
# Other inputs are optional – the last two define
# that we will get a coordinate system with axes and not frames,
# which is default for the command implicit_plot.

Page 285

var('y')
implicit_plot((x^2+y^2)^2==4*(x^2-y^2), (x,-2.5,2.5),(y,-1,1), aspect_ratio = 1,   gridlines = 'minor')+implicit_plot(x^2+y^2==1, (x,-2.5,2.5),(y,-1,1), axes = true,frame = false, color = 'red')